PETER PAZMANY SEMMELWEIS UNIVERSITYCATHOLIC UNIVERSITY Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework** Consortium leader PETER PAZMANY CATHOLIC UNIVERSITY Consortium members SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund *** **Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben * * * A p r o j e kt a z Eu r ó p a i U n i ó t á m o g a t á sá v a l , a z Eu ró p a i S z o ci á l i s Al a p t á rsf i n a n sz í ro z á sá v a l v a l ó s u l m e g . INTRODUCTION TO BIOPHYSICS (Bevezetés a biofizikába) THERMODYNAMICS (Termodinamika ) GYÖRFFY DÁNIEL, ZÁVODSZKY PÉTER . phenomenological thermodynamics describesmacroscopic properties of physical systems . rudiments of phenomenological thermodynamics . materials can be characterized by thermodynamic variables . thermodynamic state functions only depend on the state itself rather than the routes toward it . state functions have an extremum at the equilibrium of the system . the first law of thermodynamics reflects the principle of energy conservation and conversion . the second law of thermodynamics reflects theprinciple of maximal multiplicity . free energy determines the direction of reactions . macroscopic properties of a thermodynamic system can be derived from atomic structureof it . the Boltzmann-equation of entropy connects the classical and statistical thermodynamics . physical behaviour of biologicalmacromolecules can also be described bystatistical thermodynamics Phenomenological thermodynamics . phenomenological thermodynamics studiesprocesses occurs in macroscopic systemswhich are accompanied by heat transmissionand mechanical work . a physical system can be characterized bythermodynamic variables: – temperature – pressure – volume – amount of substance – in equilibrium, if the amount of substance isconstant the state of the system can be accurately characterized by only two thermodynamic variables . the ideal gas law describes the relationship among variables: pV =nRT . where p is the pressure, V is the volume, n is the number of moles, T is the temperature and R is the gas constant . DEF thermodynamic system: it is under consideration . DEF environment: which surrounds the system . DEF open system: it can exchange both matter and heat with its environment . DEF closed system: it can exchange only heatwith its environment . DEF isolated system: it can exchange neither matter nor heat with its environment . DEF thermodynamic variables: Measures that characterize the state of a macroscopicsystem. Their values only depend on the stateitself regardless of the route through which thesystem reaches it. Mathematically, they are exact differentials. They can be extensive or intensive properties. . DEF extensive properties: Their value is proportional to the amount of substance. Theyare additive. . DEF intensive properties: Their value does notdepend on the amount of substance. They arescale-invariant. . DEF state functions: They are functions ofthermodynamic variables, the value of whichdepends only on the state at which the systemstays. They have an extremum at theequilibrium of the system. Such functions are: internal energy (U), enthalpy (H), entropy (S), Helmholtz free energy (A), Gibbs free energy (G) . DEF process functions: They are functions thatdescribe a process carrying the system fromone equilibrium to another one. Such functions are heat and work. First Law of thermodynamics . in mathematical form: dU =. Q.. W where dU is an infinitesimal change of internal energy, .Q is the heat and .W is the work . This law claims that the energy can change due to heat transmission and work and these measures can be converted into each other. Figure 1. . If we move the piston by dx (figure 1.) suchthat the system is in equilibrium at any time, then the elementary work we perform is: . W =- pAdx=- pdV where .W is the elementary work, p is the pressure of gas in the piston, A is the surface area of the piston and hence dV is an infinitesimal change of the volume of the gas . the total work performed through transition ofthe system from p1,V1state to p2,V2state is: V 2 W =-. pdV V 1 Enthalpy . the first law if we consider only the work -pdV is: . Q =dU. pdV where .Q is the elementary heat flowing into or from the system and dU is the elementarychange of internal energy of the system . so at constant volume every heat we transfer increases the internal energy . DEF enthalpy: A state function. Mathematically the enthalpy is: H =U . pV . so the first law expressed with enthalpy: . Q =dH-Vdp . at constant pressure (dp=0) every heat wetransfer will increase enthalpy: . Q =dH . if we warm a body its temperature grows and ifwe cool it its temperature drops . if no work is performed then the system gainsor loses heat proportional to the change of temperature .Q .. T where .Q is the total heat flowing into or from the system and .T is the change of temperature capacity . Q C= . T where C is the heat capacity, .Q is the elementary heat flowing into or from the systemand .T is the elementary change of temperature . if internal energy is the function of temperature and volume, then: . U .U dQ=dV .dT ..V .T ..T .V . at constant volume (dV=0) the heat capacity is: ..T .V .U CV = . at constant pressure (dp=0) the heat capacity is: . H C = p ..T .p Types of thermodynamic processes . DEF isobaric process: process occurring at constant pressure . DEF isothermal process: process occurring at constant temperature . DEF isochoric process: process occurring atconstant volume . DEF adiabatic process: process with no energytransfer Figure 2. . Carnot cycle consists of four steps: 1. isothermal step from p1,V1 to p2,V2 state at T1 temperature where the system absorbs Q1 heat 2. adiabatic step from p2,V2 to p3,V3 statewhile temperature drops from T1 to T2 3.isothermal step from p3,V3 to p4,V4 state at T2 temperature where the system loses Q2 heat 4. adiabatic step from p4,V4 to p1,V1 state while temperature increases from T2 to T1 . it can be verified that for a reversible Carnot cycle: Q1 Q2 .=0 T 1 T 2 Figure 3. . any reversible cycle on the p-V diagram can beapproximated by several reversible Carnotcycles (Figure 3) for which: Qi .=0 i Ti . if approximation is infinitely fine the sum transforms to an integral: . dQ =0 T . DEF entropy (thermodynamic): a state function mathematically expressed: . Q dS = T where dS is the elementary change of entropy ofthe system, .Q is the heat flowing into or fromthe system and T is the temperature (1796-1832) (1822-1888) Second law of thermodynamics . heat can flow only from a warmer place to acooler one . in an isolated system, the entropy never decreases in a spontaneous process: dS .0 . equality sign applies to a reversible while >applies to an irreversible process . as a consequence of the first and the secondlaws of thermodynamics: dU .T·dS - p·dV . if we consider not only the pressure-volumework the equation above is: f dU -T·dS . p·dV -. X kd .k .0 k =1 . where Xk is a generalized force and d.k the corresponding generalized coordinate 1.in the case of an isolated system where dU=0, dV=0 and d.k=0: dS .0 so in equilibrium, the entropy is maximal 2.in the case of isothermal processes weintroduce a new thermodynamic variablecalled Helmholtz free energy (F) F =U -TS dF .S·dT -.W .0 because dT=0 dF ..W so the Helmholtz free energy is the maximal work which can be gained from athermodynamic process (1821-1894) 3.in the case of isothermal processes where pressure is also constant we introduce a new thermodynamic variable called Gibbs free energy: G =H -TS energy: dG-V·dp. S·dT .0 because dT=0 and dp=0 dG.0 so in equilibrium the Gibbs free energy is minimal (1839-1903) Chemical equilibrium . let us consider a chemical reaction A . B . the equilibrium constant is: [B ]eq K = [ A]eq where [A]and [B]denote the equilibrium eq eq concentration of the corresponding reactant orproduct . the free energy change of a reaction is: [ B ] .Gr =.G°r .RT ln [ A] where .G° the standard Gibbs free energy change is: .G°r =-RT ln K and [B] and [A] denote the currentconcentrations rather than equilibrium concentrations [ B]eq [B ]eq .Gr = RT ln -RT ln =0 [ A][ A] eq eq . if .Gr<0 the A › B reaction occurs spontaneously . if .Gr>0 the B › A reaction occurs spontaneously . if .Gr=0 the system in equilibrium and no reaction occurs spontaneously potentials dU TdS-pdV dH TdS-Vdp dF -pdV-SdT dG Vdp-SdT Table 1 (1844-1906) Statistical thermodynamics . understanding phenomena of thermodynamicsrequires microscopic description . fundamental principle of statistical physics isthat a macroscopic state (macrostate) can be composed by several microscopic states (microstates) . a priori every microstate has the same probability . DEF density of states: the number of microstates at a given energy level Boltzmann equation . statistical thermodynamics can be connectedwith the phenomenological thermodynamicsby Boltzmann equation S =-k ln . where S is entropy, k is the Boltzmann constant and . is the density of states (this equation is valid only in the case of isolated systems) . more generally entropy is: S =-k . pi ln pi i . because in an isolated system, a priori everymicrostate have the same probability, themacrostate being composed of the largestnumber of microstates will be the most probable one . so equilibrium can be characterized by the maximum of entropy . so equilibrium state is the most probable stateof the system – it reflects the principle of maximal multiplicity . with small but not zero probability, the systemcan escape the equilibrium state whichexplains fluctuations . let us consider an isolated system consisting of a body as a closed (energy transfer is allowed) subsystem and the environment surrounding it . in the current macrostate, subsystem can be inseveral microstates but meanwhile current macrostate of the environment can be composed by several microstates as well, sothe density of states the whole system is: .total =.subsystem·.environment . the probability of a given i state with energy Ei according to the Boltzmann distribution is: 1 Ei pi . Ei .= Ze - kT where Ei Z =. e - kT i the partition function Statistical interpretation of free energy . free energy of a macrostate of the wholesystem is: F =-kT ln Z where .total - Ei kT Z =. e i=1 . in system interacting thermally with its environment (closed system), equilibrium is characterized by the minimum of free energy(Figure 5) . because of the equation above the partitionfunction has its maximum value at equilibrium (Figure 6) Figure 5 Figure 6 Figure 7 . in a biological example, let a protein (red in figure 7) and water surrounding the protein (visible with the help of magnifying glass in figure 7) be the subsystem . let an environment be which surrounds the protein-water subsystem . let us consider the whole system as isolated so neither material nor energy transport isallowed . let degrees of freedom determining the microstate of the protein be the Cartesian coordinates of its atoms . let degrees of freedom determining themicrostate of water be the Cartesian coordinates of atoms forming water molecules . both energy of the protein and energy of the water are a function of degrees of freedom . the free energy of a given microstate of protein is determined by four factors: – energy of the protein conformation and water conformations – number of microstates that water molecules can attain, that is the entropy of solvent and theconformational entropy of the protein E protein .df p1,df p2 ... df pn. S environment .df e1 ,df e2 ...df en.